Over fierce objections from Democrats, a key House panel Wednesday passed the third in a series of five reform bills aiming to lift restrictions on how school districts and states can use federal money.

But after the party-line vote, Rep. George Miller, California Democrat and his party’s ranking member on the House Committee on Education and the Workforce, threatened “trench warfare” from this point forward.

“Allowing school districts to use federal dollars intended for poor students for something other than serving those students is morally reprehensible,” Mr. Miller said. “We won’t stand for a back-door dismantling of the federal role in education.”

Mr. Miller and fellow Democrats think the flexibility measure could widen the gap between the best- and worst-performing students. Rep. Ruben Hinojosa, Texas Democrat, called it “an attack on … civil rights” of disadvantaged students.

By giving superintendents and school boards greater authority, Republicans argue, the bill will help, rather than hurt, low-income and minority students.

Rep. Larry Bucshon, Indiana Republican, called it “insulting” for Democrats to accuse local and state education officials of wanting to use their new freedoms to chip away at programs geared toward minority students or low-income children.

For example, the pool of federal money designated for English learning is of little use to districts with few students who don’t speak the language, Republicans say. Under current law, it can’t be spent on anything else and often goes unused. By giving schools freedom to use that money for other initiatives, Republicans argue that districts can implement their own innovative ideas and better cater to their unique student bodies.

“This does not mean states and school districts will be given carte blanche to spend taxpayer dollars without any accountability,” said committee Chairman John Kline, Minnesota Republican, adding that schools would still be required to submit annual reports to the Department of Education, detailing how every dollar was spent and justifying their reasons for moving money from one pot to another.

But Democrats still fear a lack of accountability.

“Where do you think that money will go?” said Rep. Rush Holt, New Jersey Democrat. “If you leave it to what you might call the market, the privileged will get more. We have serious divisions in our society. We have serious inequalities in our society. It is incumbent on us to do everything in our power to address those.”

Democrats tried to gut the bill with six amendments, each designed to maintain tight restrictions on specific pools of money. Each was shot down on party-line votes.

The tense, three-hour back-and-forth makes it all but certain that Republicans’ reform agenda will get little, if any, support from across the aisle. Mr. Miller said the “working relationship” between the two sides will greatly suffer as a result of Wednesday’s action.

Mr. Kline has said he expects similar resistance from Democrats when the final two reform bills are introduced later this year. While the details haven’t yet been made public, the fourth bill will aim to change the definition of “effective teachers,” and the fifth and final bill will reform the accountability process for schools across the country and replace the “adequate yearly progress” system put in place as part of the 2001 No Child Left Behind Act, according to Mr. Kline.

If the House and Senate can’t come together and pass legislation soon, Education Secretary Arne Duncan has threatened to bypass Congress and offer waivers from NCLB mandates to states that demonstrate progress and a workable reform model.